# The mean of the commute time to work for a resident of a certain city is 28.3 minutes. Assume that the standard deviation of the commute time is 7.6 minutes to complete parts​ (a) through​ (c).a). what minimum percentage of commuter in the city has a commute time within 2 standard deviations of the mean?

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The mean of the commute time to work for a resident of a certain city is 28.3 minutes. Assume that the standard deviation of the commute time is 7.6 minutes to complete parts​ (a) through​ (c).

a). what minimum percentage of commuter in the city has a commute time within 2 standard deviations of the mean?

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Step 1

A normal distribution can be visualized by looking at the curve below.

X-axis shows Z score which is the distance from mean in terms of standard deviation. A z score of 2 would mean that the distance of the data point from the mean is equal to 2 standard deviation.

Commute time within 2 standard deviations of the mean would mean all communte times between Z score of -2 and +2 as shown by the shaded portion under the curve.

Step 2

The area under the curve represents probability or percentage or proportion. Since the value of probability ranges from 0 to1, the total area under the curve is always equal to 1 or 100%.

To calculate area percentage or probability corresponding to any Z score , Z tables can be used which yields percentage of area to the left of a score

We are interested in P(-2<Z<2) which will be equal to P(Z<2) - P(Z<-2)

This can be represented graphically as shown below.

Step 3

Let's see how Z table can be read for a Z score of -2 and +2

To find P(Z<=-2) , we have to find the value that lies at the intersection of row -2 and column .00

(NOTE: If this was P(Z<=-1.57), then we would look for value at the intersec... help_outlineImage Transcriptionclose00 .01 .02 .03 .04 .05 .06 .07 .08 .09 Z. 0003 .0003 0002 -3.4 0003 0003 .0003 0003 0003 0003 0003 0005 .0004 0003 3.3 0005 0005 0004 .0004 0004 0004 .0004 0005 -3.2 .0007 0006 .0006 .0006 0006 .0005 0005 0007 .0006 0007 0009 0008 0008 .0008 .0007 3.1 0010 0009 0013 0009 0008 0013 0012 0010 .0013 0012 .0011 0011 .0011 .0010 -3.0 0015 .0015 0014 0014 2.9 0019 0018 0018 0017 .0016 0016 0021 .0029 -2.8 0025 0024 0023 .0021 0026 0023 0022 0020 0019 0030 -2.7 .0028 0027 .0026 0035 0034 0033 0032 0031 0041 -2.6 0044 0039 .0047 0045 0043 0040 .0038 0037 0036 .0049 0057 .0055 0054 0052 .0051 0048 -2.5 0062 0060 0059 -2.4 0071 0082 .0080 0078 0075 0073 0069 .0068 .0066 .0064 0091 0099 .0096 0094 .0089 .0087 0084 2.3 0107 .0104 0102 -2.2 0132 .0113 0110 .0139 .0136 0129 .0125 .0122 0119 0116 0179 -2.0 .0228 0174 .0166 .0162 .0158 0154 .0150 0146 0143 2.1 0170 .0192 0188 0183 0222 0217 .0212 0207 0202 0197 0274 0268 .0262 0256 0250 0239 0233 0287 0281 .0244 -1.9 0344 .0314 0307 0301 0294 -1.8 0359 0351 0336 0329 0322 0367 -1.7 0446 0436 0427 0418 0409 0401 0392 0384 0375 0495 0485 0475 0465 0455 -1.6 0548 0537 0526 0516 0505 fullscreen

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